Grothendieck space

In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space $X$ in which every sequence in its continuous dual space $X^{\prime }$ that converges in the weak-* topology $\sigma \left(X^{\prime },X\right)$ (also known as the topology of pointwise convergence) will also converge when $X^{\prime }$ is endowed with $\sigma \left(X^{\prime },X^{\prime \prime }\right),$ which is the weak topology induced on $X^{\prime }$ by its bidual. Said differently, a Grothendieck space is a Banach space for which a sequence in its dual space converges weak-* if and only if it converges weakly.

Let $X$ be a Banach space. Then the following conditions are equivalent:

$X$ is a Grothendieck space,
for every separable Banach space $Y,$ every bounded linear operator from $X$ to $Y$ is weakly compact, that is, the image of a bounded subset of $X$ is a weakly compact subset of $Y.$
for every weakly compactly generated Banach space $Y,$ every bounded linear operator from $X$ to $Y$ is weakly compact.
every weak*-continuous function on the dual $X^{\prime }$ is weakly Riemann integrable.

Examples

Every reflexive Banach space is a Grothendieck space. Conversely, it is a consequence of the Eberlein–Šmulian theorem that a separable Grothendieck space $X$ must be reflexive, since the identity from $X\to X$ is weakly compact in this case.
Grothendieck spaces which are not reflexive include the space $C(K)$ of all continuous functions on a Stonean compact space $K,$ and the space $L^{\infty }(\mu )$ for a positive measure $\mu$ (a Stonean compact space is a Hausdorff compact space in which the closure of every open set is open).
Jean Bourgain proved that the space $H^{\infty }$ of bounded holomorphic functions on the disk is a Grothendieck space.^[1]

References

↑ J. Bourgain, $H^{\infty }$ is a Grothendieck space, Studia Math., 75 (1983), 193–216.

J. Diestel, Geometry of Banach spaces, Selected Topics, Springer, 1975.
J. Diestel, J. J. Uhl: Vector measures. Providence, R.I.: American Mathematical Society, 1977. ISBN 978-0-8218-1515-1.
Shaw, S.-Y. (2001) [1994], "Grothendieck space", Encyclopedia of Mathematics, EMS Press
Khurana, Surjit Singh (1991). "Grothendieck spaces, II". Journal of Mathematical Analysis and Applications. 159 (1). Elsevier BV: 202–207. doi:10.1016/0022-247x(91)90230-w. ISSN 0022-247X.
Nisar A. Lone, on weak Riemann integrability of weak* - continuous functions. Mediterranean journal of Mathematics, 2017.

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Grothendieck space

Examples

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